SAN DIEGO (CNN) — A wildfire scorching Southern California prompted authorities to urge thousands to evacuate on Tuesday, though authorities appear to have gotten a better grip on the blaze as the day wore on.

“At the point that the fire is right now, we have a pretty good handle on it,” San Diego Fire Chief Javier Mainar told reporters early Tuesday evening. “We hope to do some more work through the night and into tomorrow, but I think that the largest part of the emergency has passed.”

At one point late Tuesday afternoon, San Diego County announced on Twitter and its emergency department website that “more than 20,000 homes and residences are being evacuated due to the Bernardo Fire.”

But in a news conference Tuesday evening, County Sheriff Bill Gore that only 5,000 evacuation notices had been issued; he said that the 20,000 number may have come from an assumption of four people in each residence.

Authorities initially said that most of the evacuations were in San Diego city limits. But by the evening, San Diego Police Chief Shelley Zimmerman estimated about 300 homes were still evacuated in select areas.

Earlier, parents were told to pick up their children from three elementary schools because of the fire and an evacuation center was moved from one high school to another. Mainar, though, said he didn’t see any reason now why classes in those elementary schools would not resume Wednesday.

The fire began at 10:45 a.m. (1:45 p.m. ET), fanned by strong winds, according to the fire chief.

“It has been, to say the least, a very challenging day for local fire agencies and law enforcement agencies,” Mainar said. “It is unusual in May to have wind driven fires like this that prove to be such a challenge to contain.”

For all those challenges, authorities said they don’t know of any injuries or fatalities tied to the blaze. It had burned 800 acres and was 5% contained as of 7:50 p.m., according to CalFire.

The National Weather Service issued a red flag warning for areas around San Diego through 8 p.m. Wednesday. As the agency noted, “a red flag warning means that critical fire weather conditions are either occurring now or will shortly,” with strong winds, low humidity and warm temperatures feeding into “extreme fire behavior.”

Temperatures in the Southern California city are forecast to peak in the mid-90s Wednesday and Thursday with no sign of rain, before cooling somewhat later in the week.